Energy Preserving Mixed Finite Element Methods For Maxwell's Equations With Nonlinear Conductivity

Maxwell's equations are proposed by the British physicist James Maxwell to unify the laws of the electric field and magnetic field.They describe the relationships between electric field,magnetic field,charge density and current density,and are widely used in the fields of acoustics,microelectronics,optical fiber communication,etc.In this thesis,a class of energy-preserving mixed finite element method for Maxwell's equations with nonlinear conductivity is studied.For the first-order form of the nonlinear Maxwell equations,selecting appropriate edge element and face element spaces to discrete the electric field and the magnetic field respectively,and the energy-preserving semi-discrete scheme is obtained.Then using the appropriate second-order continuous-time Galerkin method(CTG)to discretize the time variable in the semi-discrete format,the energypreserving fully discrete scheme is obtained.The semi-discrete and fully discrete schemes in this paper can preserve the magnetic Gauss law exactly and have the optimal convergence order.Numerical experiments verify the correctness of the theoretical results.